Physical Meteorology ESCI 340
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Date Created: 10/15/15
ESCI 340 Physical Meteorology Cloud Physics Lesson 4 Precipitation Growth by CollisionCoalescence References A Short Course in Cloud Physics 3ml ed R0 ers and Yau Microphysic of Clouds and Precipitation 2quot ed Pruppacher and Klett Reading Rogers and Yau Chapters 8 TERMINAL VELOCITY The vertical forces on a falling droplet are it 0 Friction Fr Er2142 0 CD where CD is the drag coef cient 0 Gravity Fg 7rr3gpL Eventually the droplet will reach terminal velocity at which point the two forces are in balance This results in 142 3 0CD If we knew the drag coefficient then we could find the terminal velocity The drag coefficient is determined experimentally with the following results obtained for the terminal velocity of droplets Droplet Radius Terminal Velocity r lt 40pm u 119x106 cm ls l r2 u 8x103 s391r u 22x103 cml Zs lxpop 112 40um lt r lt 06mm r gt 06mm where p0 is equal to 120 kgm3 GROWTH DUE TO COLLISION WITH SMALLER DROPLETS OF UNIFORM SIZE As a droplet falls it may collide with other droplets The effective volume swept out by a droplet of radius R falling through a population of smaller droplets of radius r in an infinitesimally small amount of time It is dV 7rR r2 uR urdt The total mass of the smaller droplets in the volume is dm MdV 7rR r2M uR urdt where M is the liquid water content of the cloud mass of liquid water per volume of air If all the smaller droplets within the volume coalesce stick to the falling droplet then dm would also represents the increase in mass of the falling droplet so we could write dm 7 7rR r2MuR ur dt 1 CORRECTION DUE TO COLLECTION EFFICIENCY In reality not all the smaller droplets will collide with or stick coalesce to the larger droplet One reason for this is that even though a droplet may be in the path of the larger drop the air ow around the larger drop may force the smaller droplet away as the drops pass Think of a bug heading for your windshield Sometimes the bug is forced up and over your car by the air ow and is spared It is also possible for droplets that weren t in the path of the larger droplet to be captured by the wake of the falling droplet This is known as wake capture Even if two droplets collide they may not actually stick or coalesce 0 One reason for this is that there may be a microfilm of air between the droplets that prevents their surfaces from contacting All the above effects are taken into account by defining a collection ef ciency E that is just the ratio of the effective cross section of the falling drop to its actual geometric cross section The collection efficiency must be experimentally determined and is going to depend on many physical factors such as the drop sizes involved temperature turbulence wind shear etc 0 In general the collection efficiency is less than unity With collection efficiency taken into account equation 1 for the growth of droplet mass is dm E 7IEM R r2 uR ur 2 0 Equation 2 is the growth equation in terms of mass 0 Equation 2 is sometimes written as L quot KRrM 3 dt where KR r 7rER r2 uR ur 4 is called the gravitational collection kernel GROWTH EQUATION IN TERMS OF RADIUS O In terms of radius the growth equation is dR EM R r 2 7 2uRgt urgt 5 dt 4pLR 0 If we assume the smaller droplets have a negligible terminal velocity compared to the larger droplets then we can use the approximations uR ur E uR R r E R to write the growth equation in terms of radius as dR EM 7 uR 6 dt 40L From the chain rule this can be written as djdj EM uR 7 dz d dz 40L U uR where U is the velocity of any updrafts present If the updraft velocity is small this reduces to dR EM 7 E 7 8 dz 40 L This equation suggests that the droplet radius should decrease with height which makes senseas the droplet falls 1 getting smaller its radius should be increasing through the collisioncoalescence process GROWTH DUE TO COLLISION WITH SMALLER DROPLETS OF NONUNIFORM SIZE The equations we ve derived up to now apply only to a large droplet falling through a population of smaller droplets having a uniform size In reality the smaller droplets will not just have a single radius If there are two different sizes of smaller drops r1 and r2 then the growth rate in terms of mass will have a separate term for each droplet size present and will be dm 2 2 E 7rE1M1R r1 uRgt ur1gt7rE2M2 R r2 um mm where M1 and M2 are the liquid water contents for drops having radius r1 and r2 respectively and E1 and E2 are the collection efficiencies for drops having radius r1 and r2 respectively In general if there are N different sizes of little drops then we have Lm 7 Z EiMl R 02 Mao mg dt 1 le where each drop radius r has its own collection efficiency and liquid water content E and Mi Written in terms of the collection kernel we have dm 7 2 PM If the small droplets are distributed continuously then the summation becomes an integration dm EIKRrdM and we know that dM g pLPnd rdr so that the growth equation for a continuous droplet spectrum is dm 4 m 3 777 KRrrn rdr 9 dt 3pL gt do 0 Equation 9 must be solved numerically and is complicated by the fact that the collection kernel depends on radius of the small droplets so it must remain within the integral 0 The collection kernel is usually determined empirically If we assume that R gtgtr and uR gtgt ur then we can show see exercise 3 that and which are the same as equations 6 and 8 0 Thus regardless of whether or not our dropsize distribution is continuous or if we just have one size of drops present if we make the assumptions that R gtgt r and uR gtgt ur then equations 6 and 8 are valid in either case COMMENTS ON COLLISIONCOALESCENCE The growth rate is very sensitive to the drop size spectrum with broader spectra leading to faster growth rates 0 A broad spectrum is beneficial because it leads to more relative velocity between the drops and more chance for collision In a narrow spectrum all the drops are falling at roughly the same velocity and are less likely to collide Clouds droplets are initially formed via diffusion which leads to a narrowing rather than a broadening of the drop size spectra Computations using equation 9 with typical drop size spectra give growth rates too small to explain how precipitation sized drops form in reality To achieve growth rates comparable to that observed in natural clouds 15 minutes a stochastic process is required whereby a small population of fortunate drops happen to grow much faster than the average rate 0 As these drops grow very large 4 to 5 mm they become unstable and break apart This creates some additional large drops which can themselves start to grow through collisioncoalescence The physical cause of these stochastic phenomena is a source of study but turbulence is likely a factor O The collisioncoalescence process is often called the warmrain process since it is the only way to explain precipitation formation in clouds that remain above freezing However it can also occur in cold clouds EXERCISES 1 A droplet of radius 40um is at the base of a cloud 1 0 having a liquid water content of 15 gm3 and a steady updraft of 2 ms The terminal velocity of the drop is given by u 8x103 s39l R a What will the size of the drop be when it begins to fall b What is the maximum height that the drop will reach assume the collection efficiency is 1 2 a The terminal velocity uR increases with increasing R The equation dR EM uR 7 implies that for small droplets IRdz will be positive dz 40L U uR while for very large droplets IRdz will be negative What is the physical explanation for this b When the terminal velocity of the droplet equals the updraft velocity the IRdz becomes infinitely large What is the physical significance of this 3 a Show that for a continuous spectrum of small drops that the growth equation in terms of the radius of the big drop is LR 1 dt 3R2 3K R r r3nd rdr b Show that if we assume that R gtgtr and uR gtgt ur that the growth rate equation for a continuous small drop spectrum becomes LR 7rEuR m r3nd rdr dt 3 3 c Show that the expression above is equal to